Approximation of convex bodies by multiple objective optimization and an application in reachable sets

Shao, Lei; Zhao, Fangyuan; Cong, Yuhao

doi:10.1080/02331934.2018.1426583

Cited by 10 publications

(17 citation statements)

References 19 publications

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“…A similar observation was made in [12] about the problem studied there. Also here, convexity is not used within the proof, so the claim would hold also for non-convex projection problems.…”

Section: Lemma 2 1 Every Feasible Point (X Y) ∈ S Is a Minimizer Of (8) 2 For The Sets Y And P[s] It Holds Y = Proj −1 [P[s]] And P[s]supporting

confidence: 78%

“…where the last equality is due to (12). Since the shifted cone {q} + P ∞ is closed, selfboundedness of (8) follows.…”

Section: Proof Of Propositionmentioning

confidence: 98%

“…The linear counterpart suggests that multi-objective optimization might be helpful also for solving the convex projection problem. Inspired by the polyhedral case of [8] and the case of convex bodies in [12], we construct a multi-objective problem with the same feasible set S and one additional dimension of the objective space. That is, we consider the following problem…”

Section: An Associated Multi-objective Convex Problemmentioning

confidence: 99%

“…A problem similar to (CP) was considered in [12], where an outer approximation Benson algorithm was adapted to solve it under some compactness assumptions. Here, we take a more theoretical approach to relate the different problem classes.…”

Section: Introductionmentioning

confidence: 99%

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Convex projection and convex multi-objective optimization

Kováčová

Rudloff

2021

J Glob Optim

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In this paper we consider a problem, called convex projection, of projecting a convex set onto a subspace. We will show that to a convex projection one can assign a particular multi-objective convex optimization problem, such that the solution to that problem also solves the convex projection (and vice versa), which is analogous to the result in the polyhedral convex case considered in Löhne and Weißing (Math Methods Oper Res 84(2):411–426, 2016). In practice, however, one can only compute approximate solutions in the (bounded or self-bounded) convex case, which solve the problem up to a given error tolerance. We will show that for approximate solutions a similar connection can be proven, but the tolerance level needs to be adjusted. That is, an approximate solution of the convex projection solves the multi-objective problem only with an increased error. Similarly, an approximate solution of the multi-objective problem solves the convex projection with an increased error. In both cases the tolerance is increased proportionally to a multiplier. These multipliers are deduced and shown to be sharp. These results allow to compute approximate solutions to a convex projection problem by computing approximate solutions to the corresponding multi-objective convex optimization problem, for which algorithms exist in the bounded case. For completeness, we will also investigate the potential generalization of the following result to the convex case. In Löhne and Weißing (Math Methods Oper Res 84(2):411–426, 2016), it has been shown for the polyhedral case, how to construct a polyhedral projection associated to any given vector linear program and how to relate their solutions. This in turn yields an equivalence between polyhedral projection, multi-objective linear programming and vector linear programming. We will show that only some parts of this result can be generalized to the convex case, and discuss the limitations.

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“…A similar observation was made in [12] about the problem studied there. Also here, convexity is not used within the proof, so the claim would hold also for non-convex projection problems.…”

Section: Lemma 2 1 Every Feasible Point (X Y) ∈ S Is a Minimizer Of (8) 2 For The Sets Y And P[s] It Holds Y = Proj −1 [P[s]] And P[s]supporting

confidence: 78%

“…where the last equality is due to (12). Since the shifted cone {q} + P ∞ is closed, selfboundedness of (8) follows.…”

Section: Proof Of Propositionmentioning

confidence: 98%

Section: An Associated Multi-objective Convex Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Convex projection and convex multi-objective optimization

Kováčová

Rudloff

2021

J Glob Optim

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“…The following multiobjective convex program is associated to (CPP) (see [1] for a polyhedral prototype of this idea, and [2] for an extension to the non-polyhedral convex case):…”

Section: Introductionmentioning

confidence: 99%

On the approximation error for approximating convex bodies using multiobjective optimization

Löhne¹,

Zhao²,

Shao

2021

J. Appl. Numer. Optim.

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A polyhedral approximation of a convex body can be calculated by solving approximately an associated multiobjective convex program (MOCP). An MOCP can be solved approximately by Benson type algorithms, which compute outer and inner polyhedral approximations of the problem's upper image. Polyhedral approximations of a convex body can be obtained from polyhedral approximations of the upper image of the associated MOCP. We provide error bounds in terms of the Hausdorff distance for the polyhedral approximations of a convex body in dependence of the stopping criterion of the primal and dual Benson type algorithms which are applied to the associated MOCP.

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